The graph of which function has an axis of symmetry at [tex]$x=-\frac{1}{4}$[/tex]?
A. [tex]$f(x)=2 x^2+x-1$[/tex]
B. [tex]$f(x)=2 x^2-x+1$[/tex]
C. [tex]$f(x)=x^2+2 x-1$[/tex]
D. [tex]$f(x)=x^2-2 x+1$[/tex]
Answer (1)
Calculate the axis of symmetry for each quadratic function using the formula x = − 2 a b .
For f ( x ) = 2 x 2 + x − 1 , the axis of symmetry is x = − 4 1 .
Compare the calculated axes of symmetry with the given axis of symmetry x = − 4 1 .
The function f ( x ) = 2 x 2 + x − 1 has the axis of symmetry x = − 4 1 , so the answer is f ( x ) = 2 x 2 + x − 1 .
Explanation
Understanding the Problem We are given four quadratic functions and we need to find the one whose graph has an axis of symmetry at x = − 4 1 . The axis of symmetry of a quadratic function f ( x ) = a x 2 + b x + c is given by the formula x = − 2 a b . We will calculate the axis of symmetry for each function and compare it to the given value.
Calculating Axis of Symmetry for the First Function For f ( x ) = 2 x 2 + x − 1 , we have a = 2 and b = 1 . The axis of symmetry is x = − 2 ( 2 ) 1 = − 4 1 .
Calculating Axis of Symmetry for the Second Function For f ( x ) = 2 x 2 − x + 1 , we have a = 2 and b = − 1 . The axis of symmetry is x = − 2 ( 2 ) − 1 = 4 1 .
Calculating Axis of Symmetry for the Third Function For f ( x ) = x 2 + 2 x − 1 , we have a = 1 and b = 2 . The axis of symmetry is x = − 2 ( 1 ) 2 = − 1.
Calculating Axis of Symmetry for the Fourth Function For f ( x ) = x 2 − 2 x + 1 , we have a = 1 and b = − 2 . The axis of symmetry is x = − 2 ( 1 ) − 2 = 1.
Finding the Matching Function Comparing the calculated axes of symmetry with the given axis of symmetry x = − 4 1 , we find that the function f ( x ) = 2 x 2 + x − 1 has the axis of symmetry at x = − 4 1 . Therefore, the graph of f ( x ) = 2 x 2 + x − 1 has an axis of symmetry at x = − 4 1 .
Examples
Understanding the axis of symmetry is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight or satellite dish, the axis of symmetry helps ensure that the light or signal is focused correctly. Similarly, in architecture, knowing the axis of symmetry of an arch or a bridge can aid in structural design and stability. By understanding quadratic functions and their properties, we can optimize designs and ensure functionality in many engineering and architectural projects.
